Question

Determine whether the integral converges or diverges. Find the value of the integral if it converges.

Answer 1

@Reaper534 can u help?

Answer 2

Let's just focus on the integral first, keeping in mind that...
\[\Large \int\limits_1^\infty x^{-\frac43}= \lim_{b\rightarrow\infty}\int\limits_1^bx^{-\frac43}\]

Answer 3

okay

Answer 4

Well, can you integrate
\[\Large \int x ^{-\frac43}=\color{red}?\]

Answer 5

yes i get -3/3sqrt(x)

Answer 6

This
\[\Large \frac{-3}{3\sqrt{x}}\]?

Answer 7

yes!

Answer 8

Or this
\[\Large \frac{-3}{\sqrt[3]x}\]

Answer 9

the second one

Answer 10

lol... it's cube root, not 3sqrt
because \(\Large3\sqrt x\) means something entirely different, ok?

Answer 11

Okay

Answer 12

Okay, so, we have...
\[\Large \int\limits_1^b x^{-\frac43}=\left.\frac{-3}{\sqrt[3]x}\right]_1^b\]

Answer 13

Can you evaluate this bit?

Answer 14

i think

Answer 15

Well then, what do you get?

Answer 16

hmm i'm working on it

Answer 17

eh...i'm getting a wrong answer :/

Answer 18

Fundamental theorem of Calculus?
\[\Large \int\limits_a^b f'(x)dx = \left.f(x)\right]_a^b = f(b)-f(a)\]

Answer 19

ohhhh okay

Answer 20

So...
\[\Large \int\limits_1^b x^{-\frac43} \ dx=\left.\frac{-3}{\sqrt[3]x}\right]_1^b=\color{red}?\]

Answer 21

so i plug in b and 1 into that then subtract?

Answer 22

Yes... but that's not the end yet, just plug in for now, and tell me what you get :)

Answer 23

okay so -3/3sqrt(b) - -3/3sqrt(1)

Answer 24

I'm assuming by 3sqrt you mean cube root :D Okay, that being the case, you're right :)

\[\Large \int\limits_1^b x^{-\frac43}=\frac{-3}{\sqrt[3]b}+\frac3{\sqrt[3]1}= -\frac3{\sqrt[3]b}+3\]

Catch me so far?

Answer 25

yes

Answer 26

Now, we're supposed to take the improper integral to infinity, right? Remember this...
\[\Large \int\limits_1^\infty x^{-\frac43} \ dx = \color{red}{\lim_{b\rightarrow\infty}}\int\limits_1^bx^{-\frac43} \ dx\]

Now is the time to apply that limit (which we haven't done yet)
\[\Large \color{red}{\lim_{b\rightarrow\infty}}\left(-\frac3{\sqrt[3]b}+3\right) \]

Answer 27

okay

Answer 28

So... evaluating the limit...?

Answer 29

umm i get 3? :/

Answer 30

That's right!

Answer 31

ohh okay lol

Answer 32

so thats the final answer?

Answer 33

Well, you technically have two questions, but since there was an answer, then the integral converges, and it converges to 3 ^.^

Answer 34

okay thanks

Answer 35

No problem :)